Complex numbers and roots · Nombres complexes et racines
| English | Français |
|---|---|
| complex conjugate/ˈkɒmpleks ˈkɒndʒuːɡeɪt/ | conjugué complexe |
What if the root is not real?
- The equation x²+1=0 has no real roots. Extending the number system lets us represent and calculate with both roots.
- This lesson studies complex conjugate 共轭复数: The number obtained by reversing the sign of the imaginary part.
Choose the mathematical structure
- Write z=a+bi with i²=-1. The conjugate is a-bi and z times its conjugate=a²+b². The modulus is √(a²+b²); the argument needs the correct quadrant. Conjugate roots occur for polynomials with real coefficients.
- State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.
Which description correctly defines complex conjugate? · Quelle description définit correctement le conjugué complexe ?
The number obtained by reversing the sign of the imaginary part. · Le nombre obtenu en inversant le signe de la partie imaginaire.
Work through a checked case
- Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.
For z=3+4i, |z|=5 and z conjugate=3-4i. The reciprocal is (3-4i)/25. For x²-6x+13=0, the roots are 3±2i. Multiplying by the conjugate makes a complex denominator real.
Complex numbers and roots · Nombres complexes et racines
Write z=a+bi with i²=-1 · Écrivez z=a+bi avec i²=-1
Compare the model with the worked case and explain one change. · Compare le modèle avec l'exemple résolu et explique un changement.
Find |3+4i|. · Trouvez |3+4i|.
Modulus=√(3²+4²)=5. · Module=√(3²+4²)=5.
Test a tempting shortcut
- The square root of a sum is not the sum of the square roots. An inverse tangent alone may return the wrong quadrant. Do not confuse modulus with squared modulus.
- When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.
For a+bi, its modulus is always a+b. This claim is false. Explain which definition or assumption it violates.
Find the real part of (3+4i)(3-4i). · Trouvez la partie réelle de (3+4i)(3-4i).
Conjugate multiplication gives 3²+4²=25, with imaginary parts cancelling. · La multiplication par le conjugué donne 3²+4²=25, avec annulation des parties imaginaires.
For a+bi, its modulus is always a+b. · Pour a+bi, son module est toujours a+b.
The square root of a sum is not the sum of the square roots. An inverse tangent alone may return the wrong quadrant. Do not confuse modulus with squared modulus. · La racine carrée d'une somme n'est pas la somme des racines carrées. Une tangente inverse seule peut renvoyer au mauvais quadrant. Ne confondez pas le module avec le module au carré.
Interpret a new situation
- Draw the real axis horizontally and imaginary axis vertically. For products, moduli multiply and arguments add; powers extend this geometric pattern through de Moivre's theorem.
- A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.
Find the real part of 1/(3+4i). · Trouver la partie réelle de 1/(3+4i).
Multiply numerator and denominator by 3-4i: 1/(3+4i)=(3-4i)/25. Its real part is 3/25=0.12. · Multipliez le numérateur et le dénominateur par 3-4i : 1/(3+4i)=(3-4i)/25. Sa partie réelle est 3/25=0.12.
Match each part of a complete solution to its purpose. · Associer chaque partie d'une solution complète à son but.
An assumption justifies the model; a check tests the result; interpretation connects it to the question. · Une hypothèse justifie le modèle ; une vérification teste le résultat ; l'interprétation le relie à la question.
Use this in your course
- edexcel IAL pure mathematics; official unit FP2. Other-unit enrichment is identified in the scope review; it is not an extra cash-in requirement.
- Give the method before the final answer, and use the paper's calculator and formula rules. Review a wrong answer by locating the first invalid step.
The number obtained by reversing the sign of the imaginary part. Choose the relationship, show the method, check its assumptions and interpret the result.